Optimal. Leaf size=72 \[ -\frac {2 b \log (\tan (c+d x))}{a^3 d}+\frac {2 b \log (a+b \tan (c+d x))}{a^3 d}-\frac {b}{a^2 d (a+b \tan (c+d x))}-\frac {\cot (c+d x)}{a^2 d} \]
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Rubi [A] time = 0.07, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3516, 44} \[ -\frac {b}{a^2 d (a+b \tan (c+d x))}-\frac {2 b \log (\tan (c+d x))}{a^3 d}+\frac {2 b \log (a+b \tan (c+d x))}{a^3 d}-\frac {\cot (c+d x)}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 44
Rule 3516
Rubi steps
\begin {align*} \int \frac {\csc ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {1}{x^2 (a+x)^2} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac {b \operatorname {Subst}\left (\int \left (\frac {1}{a^2 x^2}-\frac {2}{a^3 x}+\frac {1}{a^2 (a+x)^2}+\frac {2}{a^3 (a+x)}\right ) \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac {\cot (c+d x)}{a^2 d}-\frac {2 b \log (\tan (c+d x))}{a^3 d}+\frac {2 b \log (a+b \tan (c+d x))}{a^3 d}-\frac {b}{a^2 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 109, normalized size = 1.51 \[ \frac {-a^2 \cot ^2(c+d x)+b^2 (2 \log (a \cos (c+d x)+b \sin (c+d x))-2 \log (\sin (c+d x))+1)-a b \cot (c+d x) (-2 \log (a \cos (c+d x)+b \sin (c+d x))+2 \log (\sin (c+d x))+1)}{a^3 d (a \cot (c+d x)+b)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 293, normalized size = 4.07 \[ -\frac {a^{2} b^{2} - {\left (a^{4} + 2 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} - {\left (a^{3} b + 2 \, a b^{3}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} b^{2} + b^{4} - {\left (a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{3} b + a b^{3}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - {\left (a^{2} b^{2} + b^{4} - {\left (a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{3} b + a b^{3}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{4} \, \cos \left (d x + c\right )^{2} + \frac {1}{4}\right )}{{\left (a^{5} b + a^{3} b^{3}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{6} + a^{4} b^{2}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - {\left (a^{5} b + a^{3} b^{3}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.89, size = 74, normalized size = 1.03 \[ \frac {\frac {2 \, b \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{3}} - \frac {2 \, b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac {2 \, b \tan \left (d x + c\right ) + a}{{\left (b \tan \left (d x + c\right )^{2} + a \tan \left (d x + c\right )\right )} a^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.47, size = 75, normalized size = 1.04 \[ -\frac {b}{a^{2} d \left (a +b \tan \left (d x +c \right )\right )}+\frac {2 b \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{3} d}-\frac {1}{d \,a^{2} \tan \left (d x +c \right )}-\frac {2 b \ln \left (\tan \left (d x +c \right )\right )}{a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 74, normalized size = 1.03 \[ -\frac {\frac {2 \, b \tan \left (d x + c\right ) + a}{a^{2} b \tan \left (d x + c\right )^{2} + a^{3} \tan \left (d x + c\right )} - \frac {2 \, b \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{3}} + \frac {2 \, b \log \left (\tan \left (d x + c\right )\right )}{a^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.84, size = 79, normalized size = 1.10 \[ \frac {2\,b\,\ln \left (\frac {a+b\,\mathrm {tan}\left (c+d\,x\right )}{\mathrm {tan}\left (c+d\,x\right )}\right )}{a^3\,d}-\frac {2\,b}{a^2\,d\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}-\frac {1}{a\,d\,\mathrm {tan}\left (c+d\,x\right )\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{2}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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